Why isn't this relation transitive?

Question

Here's a question from my e-text:

By definition, a relation $R$ on $A$ is transitive if whenever $xRy$ and $yRz$, then $xRz$. That is, the logic is sort of like a "chain" or a hypothetical syllogism, if you will.

So in the diagram above, you have $xQy$ and $yQx$, and you also have $xQx$, so why isn't this transitive? I don't quite understand their logic.

Answer 1

That is only one case. You have to check that for all elements $a,b,c$ in the set on which the relation is being made, if it is true that $aRb$ and $bRc$, then it should also be true that $ aRc$. You have checked this for only one possible assignment of $a,b,c$, namely $a=c=x$ and $b=y$. You should also check what happens when $a=c=y$ and $b=x$, which is another possibility. This translates to : If $yRx$ and $xRy$ then $yRy$, which is not true here. Hence, the relation is not transitive.

Moral of the story : transitivity must be checked for all possible values of $a,b,c$. That is to say, for all triples of elements $(a,b,c)$ from the set, whenever it happens that $aRb$ and $aRc$, it must happen that $aRc$, but that is not the case above for all possibilities, although it is true in one case (the one you pointed out). So it's really the difference between one and all.